What is the formula for the area of a circle?

The formula for the area of a circle is A = πr^2, where A represents the area, π is a constant approximately equal to 3.14159, and r stands for the radius of the circle. This formula arises from the geometric properties of circles and represents the space contained within the circle's circumference.

The reason this formula works is rooted in the definition of area, which measures the extent of a two-dimensional space. The radius is the distance from the center of the circle to any point on its circumference. When you square the radius (r^2), you determine the proportion of the area relative to the unit squares that fit into that space. Multiplying this by π adjusts for the circular shape, essentially converting the square units to fit the curves and proportions of a circle.

The other formulas provided do not accurately represent the area. For instance, the formula involving 2(π)r calculates the circumference of a circle, not its area. The formula A = πd incorporates the diameter but fails to account for the squaring of the radius, which is necessary for area calculation. Lastly, simply equating area to r^2 lacks any measures of A's proportionality to the circular dimensions. Thus, A = π